A024023 -id:A024023 - cp's OEIS frontend

A128760 Number of ways to write n as the absolute difference of a power of 2 and a power of 3.

1, 4, 1, 1, 0, 3, 0, 3, 1, 0, 0, 1, 0, 2, 0, 1, 0, 1, 0, 1, 0, 0, 0, 2, 0, 1, 1, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0

Offset: 0

Reinhard Zumkeller, Mar 25 2007

nonn

a(A014121(n)) > 0; the only even numbers m with a(m)>0 are of the form m=3^k-1: a(A024023(n)) > 0;

Conjecture: there exists c>=23 such that a(n)<2 for n>c.

			a(1) = #{2^1 - 3^0, 2^2 - 3^1, 3^1 - 2^1, 3^2 - 2^3} = 4;
a(2) = #{3^1 - 2^0} = 1;
a(3) = #{2^2 - 3^0} = 1;
a(5) = #{2^3 - 3^1, 2^5 - 3^3, 3^2 - 2^2} = 3;
a(7) = #{2^3 - 3^0, 2^4 - 3^2, 3^2 - 2^1} = 3;
a(8) = #{3^2 - 2^0} = 1;
a(11) = #{3^3 - 2^4} = 1;
a(13) = #{2^4 - 3^1, 2^8 - 3^5} = 2;
a(15) = #{2^4 - 2^0} = 1;
a(17) = #{3^4 - 2^6} = 1;
a(19) = #{3^3 - 2^3} = 1;
a(23) = #{2^5 - 3^2, 3^3 - 2^2} = 2;
a(25) = #{3^3 - 2^1} = 1.

Max Alekseyev, Table of n, a(n) for n = 0..1000

Cf. A000079, A000244.

A238976 a(n) = ((3^(n-1)-1)^2)/4.

Original entry on oeis.org

0, 1, 16, 169, 1600, 14641, 132496, 1194649, 10758400, 96845281, 871666576, 7845176329, 70607118400, 635465659921, 5719195722256, 51472775849209, 463255025689600, 4169295360346561, 37523658630539536, 337712928837117289, 3039416363020840000, 27354747277647913201, 246192725530212278416

Offset: 1

Kival Ngaokrajang, Mar 07 2014

If the Cantor square fractal is modified as shown in the illustration (see Links), then 4*a(n) is the total number of holes in the modified Cantor square fractal after n iterations. The total number of sides (outside) is 4*A171498(n-1). The total length of the sides (outside) converges to 20 when the initial total side length is 12 (starting with 5 unit squares).
For the Cantor square fractal, the total number of sides (outside) is 4*A168616(n+2). The total number of holes is 4*A060867(n-1) for n > 1. The total length of the sides (outside) converges to 12 with the same initial condition (i.e., 5 unit square); its maximum is 17.333... and is reached at n = 2, 3. The Cantor square fractal and modified one are not true fractals.
See illustrations in links.

Kival Ngaokrajang, Illustrations of initial terms
Eric Weisstein's World of Mathematics, Cantor Square Fractal
Index entries for linear recurrences with constant coefficients, signature (13,-39,27)

Cf. A024023, A060867, A060868, A168616, A171498.

a(n) = ((3^(n-1)-1)^2)/4; \\ Joerg Arndt, Mar 08 2014

a(n) = (A024023(n-1))^2/4.

G.f.: x*(3*x + 1)/((1-x)*(1-3*x)*(1-9*x)). - Ralf Stephan, Mar 14 2014

A249435 a(1) = 0, after which one less than prime powers p^m with exponent m >= 2.

Original entry on oeis.org

0, 3, 7, 8, 15, 24, 26, 31, 48, 63, 80, 120, 124, 127, 168, 242, 255, 288, 342, 360, 511, 528, 624, 728, 840, 960, 1023, 1330, 1368, 1680, 1848, 2047, 2186, 2196, 2208, 2400, 2808, 3124, 3480, 3720, 4095, 4488, 4912, 5040, 5328, 6240, 6560, 6858, 6888, 7920, 8191, 9408, 10200, 10608, 11448

Offset: 1

Antti Karttunen, Nov 02 2014

nonn

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

One less than A025475.
Subsequence of A181062 and also a subsequence of A249433 (after the initial zero).
Union of sequences A000225, A024023, A024049, A024075, A024127, etc. without their term a(1).
Apart from the first term, subsequence of A045542.

list(lim)=my(v=List([0])); lim=lim\1+1; for(m=2,logint(lim,2), forprime(p=2,sqrtnint(lim,m), listput(v, p^m-1))); Set(v) \\ Charles R Greathouse IV, Aug 26 2015

Scheme
```
(define (A249435 n) (- (A025475 n) 1))
```

a(n) = A025475(n) - 1.

A263647 Numbers k such that 2^k-1 and 3^k-1 are coprime.

Original entry on oeis.org

1, 2, 3, 5, 7, 9, 13, 14, 15, 17, 19, 21, 25, 26, 27, 29, 31, 34, 37, 38, 39, 41, 45, 47, 49, 51, 53, 57, 59, 61, 62, 63, 65, 67, 71, 73, 74, 79, 81, 85, 87, 89, 91, 93, 94, 97, 98, 101, 103, 107, 109, 111, 113, 118, 122, 123, 125, 127, 133, 134, 135, 137, 139, 141, 142, 145, 147, 149, 151, 153, 157, 158, 159, 163, 167, 169, 171

Offset: 1

Robert Israel, Oct 22 2015

n such that there is no k for which both A014664(k) and A062117(k) divide n.
If n is in the sequence, then so is every divisor of n.
1 and 2 are the only members that are in A006093.
Conjectured to be infinite: see the Ailon and Rudnick paper.

			gcd(2^1-1, 3^1-1) = gcd(1,2) = 1, so a(1) = 1.
gcd(2^2-1, 3^2-1) = gcd(3,8) = 1, so a(2) = 2.
gcd(2^4-1, 3^4-1) = gcd(15,80) = 5, so 4 is not in the sequence.

Robert Israel, Table of n, a(n) for n = 1..10000
Nir Ailon, Zéev Rudnick, Torsion points on curves and common divisors of a^k - 1 and b^k - 1, Acta Arith. 113 (2004), 31-38. Also arXiv:math/0202102 [math.NT], 2002.
Thomas Bloom, Problem 820, Erdős Problems.

Cf. A000225, A006093, A024023, A014664, A062117.

[n: n in [1..200] | Gcd(2^n-1, 3^n-1) eq 1]; // Vincenzo Librandi, May 01 2016

select(n -> igcd(2^n-1,3^n-1)=1, [$1..1000]);

Select[Range[200], GCD[2^# - 1, 3^# - 1] == 1 &] (* Vincenzo Librandi, May 01 2016 *)

A283454 The square root of the smallest square referenced in A249025 (Numbers k such that 3^k - 1 is not squarefree).

Original entry on oeis.org

2, 2, 11, 2, 2, 2, 2, 2, 11, 2, 2, 2, 2, 2, 11, 2, 2, 2, 2, 2, 11, 2, 2, 13, 2, 2, 2, 11, 2, 2, 2, 2, 2, 11, 2, 2, 2, 2, 2, 11, 2, 2, 2, 2, 2, 11, 2, 2, 2, 2, 2, 11, 2, 2, 2, 2, 2, 11, 2, 2, 2, 2, 2, 11, 2, 2, 2, 2, 2, 11, 2, 13, 2, 2, 2, 2, 11, 2, 2, 2, 2, 2, 11

Offset: 1

Robert Price, Mar 07 2017

nonn

The terms are the smallest prime whose square divides 3^k-1, when it is not squarefree.

			A249025(3)=5, 3^5-1 = 242 = 2*11*11. 242 is not squarefree the square being 11*11 = 121, the root being 11.

Amiram Eldar, Table of n, a(n) for n = 1..407 (terms 1..121 from Michel Marcus)

Cf. A024023, A046027, A249025, A249739, A283453.

p[n_] := If[(f = Select[FactorInteger[n], Last[#] > 1 &]) == {}, 1, f[[1, 1]]]; p /@ Select[3^Range[100] - 1, !SquareFreeQ[#] &] (* Amiram Eldar, Feb 12 2021 *)

lista(nn) = {for (n=1, nn, if (!issquarefree(k = 3^n-1), f = factor(k/core(k)); vsq = select(x->((x%2) == 0), f[,2], 1); print1(f[vsq[1], 1], ", ");););} \\ Michel Marcus, Mar 11 2017

a(n) = A249739(A024023(A249025(n))). - Amiram Eldar, Feb 12 2021

More terms from Michel Marcus, Mar 11 2017

A330168 Length of the longest run of 2's in the ternary expression of n.

Original entry on oeis.org

0, 0, 1, 0, 0, 1, 1, 1, 2, 0, 0, 1, 0, 0, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 2, 3, 0, 0, 1, 0, 0, 1, 1, 1, 2, 0, 0, 1, 0, 0, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 2, 3, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 3, 3, 4, 0, 0, 1, 0, 0, 1, 1, 1, 2, 0

Offset: 0

Joshua Oliver, Dec 04 2019

All numbers appear in this sequence. Numbers of the form 3^n-1 (A024023(n)) have n 2's in their ternary expression.

The longest run of zeros possible in this sequence is 2, as the last digit of the ternary expression of the integers cycles between 0, 1, and 2, meaning that at least one of three consecutive numbers has a 2 in its ternary expression.

			For n = 74, the ternary expression of 74 is 2202. The length of the runs of 2's in the ternary expression of 74 are 2 and 1, respectively. The larger of these two values is 2, so a(74) = 2.
   n [ternary n] a(n)
   0 [        0] 0
   1 [        1] 0
   2 [        2] 1
   3 [      1 0] 0
   4 [      1 1] 0
   5 [      1 2] 1
   6 [      2 0] 1
   7 [      2 1] 1
   8 [      2 2] 2
   9 [    1 0 0] 0
  10 [    1 0 1] 0
  11 [    1 0 2] 1
  12 [    1 1 0] 0
  13 [    1 1 1] 0
  14 [    1 1 2] 1
  15 [    1 2 0] 1
  16 [    1 2 1] 1
  17 [    1 2 2] 2
  18 [    2 0 0] 1
  19 [    2 0 1] 1
  20 [    2 0 2] 1

Wikipedia, Ternary numeral system.

Cf. A007089, A024023, A081603, A330036, A330166, A330167.

Equals zero iff n is in A005836.

Table[Max@FoldList[If[#2==2,#1+1,0]&,0,IntegerDigits[n,3]],{n,0,90}]

a(A024023(n)) = a(3^n-1) = n.

a(n) = 0 iff n is in A005836.

A044972 Numbers whose base-3 representation contains exactly one 0 and two 1's.

Original entry on oeis.org

10, 12, 32, 34, 38, 42, 46, 48, 58, 64, 66, 98, 104, 106, 116, 128, 132, 140, 142, 146, 150, 154, 156, 176, 178, 184, 194, 196, 200, 204, 208, 210, 220, 226, 228, 296, 314, 320, 322, 350, 386, 398, 402, 422, 428, 430, 440, 452

Offset: 1

Clark Kimberling

David A. Corneth, Table of n, a(n) for n = 1..10000

Cf. A007089, A024023.

Select[Range[500],DigitCount[#,3,0]==1&&DigitCount[#,3,1]==2&] (* Harvey P. Dale, Aug 19 2019 *)

is(n)=my(d=digits(n,3),fr=vector(3));for(i=1,#d,fr[d[i]+1]++);fr[1]==1&&fr[2]==2 \\ David A. Corneth, Aug 19 2019

A103453 a(n) = 0^n + 3^n - 1.

Original entry on oeis.org

1, 2, 8, 26, 80, 242, 728, 2186, 6560, 19682, 59048, 177146, 531440, 1594322, 4782968, 14348906, 43046720, 129140162, 387420488, 1162261466, 3486784400, 10460353202, 31381059608, 94143178826, 282429536480, 847288609442

Offset: 0

Paul Barry, Feb 06 2005

A transform of 3^n under the matrix A103452.

a(n) is the number of moves required to solve a Towers of Hanoi puzzle of 3 towers in a line (no direct connection between the two towers on the ends) with n pieces to be moved from one end tower to the other. This is easily proved through demonstration. - Roderick Kimball, Nov 22 2015

Vincenzo Librandi, Table of n, a(n) for n = 0..190
Steven Schlicker, Roman Vasquez, and Rachel Wofford, Integer Sequences from Configurations in the Hausdorff Metric Geometry via Edge Covers of Bipartite Graphs, J. Int. Seq. (2023) Vol. 26, Art. 23.6.6.
Index entries for linear recurrences with constant coefficients, signature (4,-3).

Cf. A103452.

Essentially identical to A024023.

[0^n+3^n-1: n in [0..30] ]; // Vincenzo Librandi, Apr 30 2011

Table[If[n==0, 1, 3^n -1], {n, 0, 30}] (* G. C. Greubel, Jun 18 2021 *)
LinearRecurrence[{4,-3},{1,2,8},30] (* Harvey P. Dale, Feb 13 2022 *)

a(n) = if(n==0, 1, 3^n-1); \\ Altug Alkan, Nov 22 2015

[3^n -1 +0^n for n in (0..30)] # G. C. Greubel, Jun 18 2021

G.f.: (1 -2*x +3*x^2)/((1-x)*(1-3*x)).
a(n) = Sum_{k=0..n} A103452(n, k)*3^k.
a(n) = Sum_{k=0..n} (2*0^(n-k) - 1)*0^(k*(n-k))*3^k.
From G. C. Greubel, Jun 18 2021: (Start)
E.g.f.: 1 - exp(x) + exp(3*x).
a(n) = [n=0] + 2*A003462(n). (End)

A114804 The numbers 3^n-1 written in groups of three digits, with leading zeros omitted.

Original entry on oeis.org

282, 680, 242, 728, 218, 665, 601, 968, 259, 48, 177, 146, 531, 440, 159, 432, 247, 829, 681, 434, 890, 643, 46, 720, 129, 140, 162, 387, 420, 488, 116, 226, 146, 634, 867, 844, 1, 46, 35, 320, 231, 381, 59, 608, 941, 431, 788, 262, 824, 295, 364, 808

Offset: 1

Jonathan Vos Post, Feb 18 2006

			2, 8, 26, 80, 242, 728, 2186, ...

Harvey P. Dale, Table of n, a(n) for n = 1..10000

Cf. A024023, A114645, A114807, A114808.

FromDigits/@Partition[Flatten[IntegerDigits/@(3^Range[30]-1)],3] (* Harvey P. Dale, Jul 04 2021 *)

Terms recomputed to use the definition equivalent to A114645. - R. J. Mathar, Jun 23 2014

A125725 Numbers whose base-7 representation is 222....2.

Original entry on oeis.org

0, 2, 16, 114, 800, 5602, 39216, 274514, 1921600, 13451202, 94158416, 659108914, 4613762400, 32296336802, 226074357616, 1582520503314, 11077643523200, 77543504662402, 542804532636816, 3799631728457714, 26597422099204000

Offset: 1

Zerinvary Lajos, Feb 02 2007

			base 7.......decimal
0..................0
2..................2
22................16
222..............114
2222.............800
22222...........5602
222222.........39216
2222222.......274514
22222222.....1921600
222222222...13451202
etc...........etc.

Davis Smith, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (8,-7).

Cf. A007310, A023000, A047522, A165759.

Cf. also A002276, A005610, A020988, A024023, A125831, A125835, A125857 for related or similarly constructed sequences.

List([1..25], n-> (7^(n-1) -1)/3); # G. C. Greubel, May 23 2019

[0] cat [n:n in [1..15000000]| Set(Intseq(n,7)) subset [2]]; // Marius A. Burtea, May 06 2019

[(7^(n-1)-1)/3: n in [1..25]]; // Marius A. Burtea, May 06 2019

Maple
```
seq(2*(7^n-1)/6, n=0..25);
```

FromDigits[#,7]&/@Table[PadLeft[{2},n,2],{n,0,25}]  (* Harvey P. Dale, Apr 13 2011 *)
(7^(Range[25]-1) - 1)/3 (* G. C. Greubel, May 23 2019 *)

vector(25, n, (7^(n-1)-1)/3) \\ Davis Smith, Apr 04 2019

[(7^(n-1) -1)/3 for n in (1..25)] # G. C. Greubel, May 23 2019

a(n) = (7^(n-1) - 1)/3 = 2*A023000(n-1).
a(n) = 7*a(n-1) + 2, with a(1)=0. - Vincenzo Librandi, Sep 30 2010
G.f.: 2*x^2 / ( (1-x)*(1-7*x) ). - R. J. Mathar, Sep 30 2013
From Davis Smith, Apr 04 2019: (Start)
A007310(a(n) + 1) = 7^(n - 1).
A047522(a(n + 1)) = -1*A165759(n). (End)
E.g.f.: (exp(7*x) - 7*exp(x) + 6)/21. - Stefano Spezia, Jan 12 2025

Offset corrected by N. J. A. Sloane, Oct 02 2010

cp's OEIS Frontend

A128760 Number of ways to write n as the absolute difference of a power of 2 and a power of 3.

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A238976 a(n) = ((3^(n-1)-1)^2)/4.

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A249435 a(1) = 0, after which one less than prime powers p^m with exponent m >= 2.

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A263647 Numbers k such that 2^k-1 and 3^k-1 are coprime.

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A283454 The square root of the smallest square referenced in A249025 (Numbers k such that 3^k - 1 is not squarefree).

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A330168 Length of the longest run of 2's in the ternary expression of n.

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Mathematica

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A044972 Numbers whose base-3 representation contains exactly one 0 and two 1's.

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Mathematica

PARI

A103453 a(n) = 0^n + 3^n - 1.

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